Flexure Revisited: Strength of Singly Reinforced Beams- A ...
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Al-Nahrain University, College of Engineering Journal (NUCEJ) Vol.18 No.1, 2015 pp.1-15
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Flexure Revisited: Strength of Singly Reinforced Beams- A
Simple Approach
Raid I. Khalel Nisreen S. Mohammed*
Sameh B.T. Shukur Kaiss F. Sarsam
Building & Constructions Department
University of Technology *E-mail: [email protected]
Abstract 181 singly reinforced beam tests available
from the literature are investigated to obtain two
flexural design methods-one simple proposal and
an alternative one that takes into account the
influence of rising flexural reinforcement ratio on
design. The simple proposal takes significantly
less time to apply compared to the alternative one.
The former leads to only 0.5% greater COV than
the latter. These design methods are compared
with four code methods (ACI 318M-11, 318M-99,
BS and NZ codes).
Based on the ratio of ( being the
calculated moment resistance), the two proposal
methods give relatively low coefficient of variation
(COV) values: 15.9% for the simple method and
15.4% for the alternative method. These compare
to COV values of 17.5%, 15.9%, 15.6% and
15.7% for the ACI 318M-11, ACI 318M-99, BS
and NZ methods, respectively.
One major advantage of the simple design
method (Constant value of = 0.85) is that all 181
tests lead to safe prediction.
Keywords: beams; flexure; nominal strength;
reinforced concrete; ultimate strength.
Introduction The most recent ACI code (318M-11) [1]
treats beams in flexure with varying values of the
strength reduction factor . This is basically
modified from the first code in 2002 [2] which
relates to the difference between tension control,
compression control and transition zone. Reference
1 relates to Fig.1.
This trend contrasts with previous ACI code
design (1999 code [3] and editions prior to 1999)
where has a constant value of 0.9. Other codes
[4,5] have a different approach to RC beam
flexural design. These codes also have simple
design approaches to flexural design of RC beams,
in a similar manner to reference 3.
Figure.1: Variation of with net tensile
strain in extreme tension steel, t, and c/dt-
ACI 318M-11 [1].
In this work it is intended to apply design in
flexure for singly reinforced beams, using
experimental data from 181 tests published in the
literature [6-28]. Table 1 indicates the ranges of
values for the variables in these beams.
Table 1: Details of 181 beam tests [6-28] Variable Range
f’c, MPa (psi) 8.6-110 (1244-15915)
ρ 0.0021-0.0684
b/d 0.425-1.531
c/d 0.031-0.684
*All specimens were tested at 28 days.
Research Significance A simplified design of RC beams in flexure is
introduced, which will be related to 181 test data
of singly reinforced beams failing in flexure. This
will be compared to other simple code design
methods [3-5]. In addition, design by ACI 318M-
11 [1] is also included in the comparison.
An alternative modification to reference 1 is
also studied and included in this work, in an effort
to reduce the COV of the ratio of tested/calculated
moment capacity.
Existing Design Methods The following includes brief details of design
based on 4 code methods:
1. ACI 318M-11 [1] Code design Essentially this design for singly reinforced
beams is based on the details of Fig.1, in addition
to Fig.2.
NUCEJ Vol.18 No.1. 2015 Khalel, et al., pp.1-15
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{
(
)
[ ]
{
[
]
Figure.2: Ultimate stress and strain
distribution at the cross section ACI 318M [1-
3].
2. ACI 318M-99 [3] Code design This design is essentially identical with
reference 1, except for using a constant value of
3. BS 8110: 1997 [4] Code design Design is based on Fig.3.
Where is used per Eq. (8):
= 0.8 fcu
(
)
4. New Zealand [5] Code design Design is based on Fig.2, except for the
modification of Eq. (10) for α1:
{
Where [ ] is per Eq. (6); using a constant
value of .
Figure.3: Ultimate stress and strain
distribution at the cross section-BS 8110 [4].
Proposed Simple Design Similarly to other simple code design
methods [3-5], Eq. (12) is proposed for flexural
design of singly reinforced concrete beams:
(
)
Eq. (12) is a result of regression analysis,
including all details, leading to the lowest possible
COV. In this equation, (a) is applied per Eq. (1),
which agrees with reference 1 (ACI 318M-11
Code).
Alternative Design Method To acknowledge the influence on ductility of
lower steel ratios, the following procedure is
proposed as an alternative design for reinforced
beams in flexure. This method uses the details
indicated in Fig.2.
Eq. (13) gives the influence of on and
{
This alternative design equation is identical
with Eq. (6), where follows Eq. (14) and the
details of Fig.4:
{
[
]
Eqs. (13) & (14) are intended to achieve the
lowest COV values for the alternative method,
based on regression analysis.
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Figure.4: Variation of with net tensile strain
in extreme tension steel, t, and c/dt for the
alternative design proposal.
Comparison of Results and Discussion The 6 methods compared in this work have
quite different approaches to predicting Mr for
singly reinforced beams. Following are some
comments on the studies made in this work:
17 of the 181 test results have a higher than
; the latter as defined by reference 1:
(
)
In all cases of comparison for these 17 beams
( , per ACI 318M-11 [1]) the strength is
based on Eq. (16):
(
)
Where:
Eq. (17) is applied to all cases when fs< f y.
Table 2 gives a comparison of the 6 methods
studied in this work: The highest COV is due to
the present ACI code [1] at 17.5 percent. This
contrasts with significantly lower COV values by
other methods, ranging between 15.4% - 15.9%.
Thus applying reference 1 design raises the COV
between 10.1 and 13.6 percent, compared to the
other methods. In contrast with the proposed
simplified method, all other methods lead to
between 3-7 unsafe predictions- the former leads
to safe predictions for all 181 tests, Table 2.
Table 2: Statistical analysis of the ratio of (Mtest /Mr) for 181 beam tests
Detail ACI-11 [1] ACI-99 [3]
BS-97 [4] NZ-9 5[5] Simple method Alternative method
1.348 1.268 1.290 1.272 1.343 1.288
S.D. 0.236 0.202 0.201 0.199 0.214 0.198
COV% 17.499 15.915 15.603 15.670 15.903 15.374
Low 0.954 0.954 0.935 0.954 1.010 0.946
High 2.326 2.315 2.204 2.319 2.451 2.319
High/Low 2.439 2.427 2.357 2.431 2.427 2.450
Number<1 4 7 3 4 0 5
Because of the significantly different
approaches in the 6 methods being applied in this
work, different failure modes are accommodated in
these methods, regardless of whether failure is in
reinforcement (tensile) or concrete (crushing).
Figs. 5-7 show a comparison between the 6
considered methods of design in flexure. Based on
the effect of on the relative safety, there is a
trend for slightly lower safety factor with rising
in all methods, except the alternative proposal for
modified ACI-11 method. In contrast to all other
methods, the drop of safety is more significant
with rising , when the BS method is used.
All methods, except for design per ACI
318M-11 [1], lead to a drop in the safety factor as
rises. Despite this difference between
reference 1 design and the proposed simplified
design method, the latter leads to safe results for
all 181 tests. This clearly stands out as an added
advantage of the proposed simplified design
method.
For the ratio of b/d between 0.425-1.531,
there seems to be no significant difference between
the 6 methods.
Conclusions Based on 181 tests of singly reinforced beams
failing in flexure, a study has been made on the
prediction of strength, based on 6 methods – 4
code methods, plus 2 alternative ones: A
simplified approach and one modified from the
latest ACI code [1]. The following conclusions are
drawn:
1. Essentially all 6 design methods are safe,
with the ratio of being less than
one between 0 – 7 cases only out of 181
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tests. The only method with no unsafe
prediction is the proposed simplified
method, where all ratios of are
greater than 1 for the 181 tests - as
evidenced in Table 2.
2. The two methods presented in this work
contrast with the present ACI code method
in their COV values. While reference 1 has
the highest COV of all 6 methods of
17.5%, the two presented methods lead to
only 15.4% and 15.9% for the COV of the
ratio of .
3. The proposed alternative modification of
reference 1 shows no effect on safety of
prediction with rising . In contrast, BS
design leads to a significant drop in safety
of prediction with rising .
4. Five of the 6 methods lead to a drop with
the safety factor as rises, in
contrast with ACI 318M-11 [1] code,
which has no drop in safety. While the
proposed design method (with fixed
) has a drop in safety of
prediction, this method leads to safe
prediction in all 181 tests.
5. With a significant range of b/d (0.425 to
1.531), all 6 methods show no change in
safety with the value of b/d.
Figure.5: Influence of compressive strength of concrete on Mtest /Mr ratio.
y = -0.0015x + 1.4019 0.5
1
1.5
2
2.5
0 30 60 90 120
Mte
st/M
r
f'c (MPa)
ACI-11
y = -0.0008x + 1.2967 0.5
1
1.5
2
2.5
0 30 60 90 120
Mte
st/M
r
ƒ'c (MPa)
ACI-99
y = -0.0021x + 1.3623 0.5
1
1.5
2
2.5
0 30 60 90 120
Mte
st/M
r
f'c (MPa)
BS-97
y = -0.0002x + 1.2794 0.5
1
1.5
2
2.5
0 30 60 90 120
Mte
st/M
r
f'c (MPa)
NZ
y = -0.0009x + 1.3732 0.5
1
1.5
2
2.5
0 30 60 90 120
Mte
st/M
r
f'c (MPa)
Simple Method
y = 0.0004x + 1.272
0.5
1
1.5
2
2.5
0 30 60 90 120
Mte
st/M
r
f'c (MPa)
Alternative Method
NUCEJ Vol.18 No.1. 2015 Khalel, et al., pp.1-15
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Figure.6: Influence of tension steel reinforcement ratio ρ on Mtest /Mr ratio.
y = 1.454x + 1.3234 0.5
1
1.5
2
2.5
0 0.02 0.04 0.06 0.08
Mte
st/M
r
ρ
ACI-11
y = -6.5191x + 1.3801 0.5
1
1.5
2
2.5
0 0.02 0.04 0.06 0.08
Mte
st/M
r
ρ
ACI-99
y = -2.8568x + 1.3389 0.5
1
1.5
2
2.5
0 0.02 0.04 0.06 0.08
Mte
st/M
r
ρ
BS-97
y = -5.6411x + 1.369 0.5
1
1.5
2
2.5
0 0.02 0.04 0.06 0.08
Mte
st/M
r
ρ
NZ
y = -6.8664x + 1.461 0.5
1
1.5
2
2.5
0 0.02 0.04 0.06 0.08
Mte
st/M
r
ρ
Simple Method
y = -2.9473x + 1.3383 0.5
1
1.5
2
2.5
0 0.02 0.04 0.06 0.08
Mte
st/M
r
ρ
Alternative Method
NUCEJ Vol.18 No.1. 2015 Khalel, et al., pp.1-15
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Figure.7: Influence of beam width to effective depth ratio b/d on Mtest /Mr ratio.
Notation a = depth of equivalent rectangular stress block,
mm
As = area of nonprestressed longitudinal tension
reinforcement, mm2
b = width of compression face of member, mm
c = distance from extreme compression fiber to
neutral axis, mm
C = compression force in concrete, N
COV = coefficient of variation of the ratio of
Mtest/Mr
d = distance from extreme compression fiber to
centroid of longitudinal tension reinforcement, mm
dt = distance from extreme compression fiber to
centroid of extreme layer of longitudinal tension
steel, mm
f’c = specified cylinder compressive strength of
concrete, MPa
fcu = specified cube compressive strength of
concrete, MPa
y = -0.0255x + 1.3675 0.5
1
1.5
2
2.5
0.4 0.8 1.2 1.6
Mte
st/M
r
b/d
ACI-11
y = -0.0575x + 1.3108 0.5
1
1.5
2
2.5
0.4 0.8 1.2 1.6
Mte
st/M
r
b/d
ACI-99
y = -0.026x + 1.3091 0.5
1
1.5
2
2.5
0.4 0.8 1.2 1.6
Mte
st/M
r
b/d
BS-97
y = -0.056x + 1.3136 0.5
1
1.5
2
2.5
0.4 0.8 1.2 1.6
Mte
st/M
r
b/d
NZ
y = -0.061x + 1.3884
0.5
1
1.5
2
2.5
0.4 0.8 1.2 1.6
Mte
st/M
r
b/d
Simple Method
y = -0.057x + 1.3301
0.5
1
1.5
2
2.5
0.4 0.8 1.2 1.6
Mte
st/M
r
b/d
Alternative Method
NUCEJ Vol.18 No.1. 2015 Khalel, et al., pp.1-15
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fs = calculated tensile stress in longitudinal tension
steel, MPa
fy = specified yield strength of reinforcement, MPa
h = overall thickness or height of member, mm
Mn = nominal flexural strength at section, N.mm
Mr = calculated moment resistance at section,
N.mm
Mtest = tested moment resistance at section, N.mm
Mu = factored moment at section, N.mm
S.D. = standard deviation of the ratio of Mtest /Mr
T = tension force in longitudinal tension steel, N
= arithmetic mean of the ratio of Mtest /Mr
α1 = factor related to , e.g. α1 = 0.85 for ACI
design [1-3]
= factor relating depth of equivalent rectangular
compression stress block to neutral axis depth
= maximum usable strain at extreme concrete
compression fiber
= net tensile strain in extreme layer of
longitudinal tension steel at nominal strength
= ratio of As to bd
ratio of As to bd producing balanced strain
Conditions [1-3]
= strength reduction factor
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طريقة مبسطة -مقاومة الانثناء للعتبات الخرسانية أحادية التسليح
قيس فؤاد سرسم سامح بدري طوبيا نسرين صالح محمد خليلرائد ابراهيم
الجامعة التكنولوجية قسم هندسة البناء والانشاءات
العراق –بغداد
الخلاصة:
عتبة خرسانية أحادية التسليح مأخوذة من بحوث سابقة وذلك للحصول على 131تم دراسة نتائج فحوصات الاولى طريقة مقترحة بسيطة، اما الثانية فتأخذ بنظر الاعتبار تأثير زيادة نسبة حديد تسليح طريقتين لتصميم الانثتاء،
الانثناء على التصميم. ان تطبيق الطريقة المبسطة يحتاج الى وقت اقصر مقارنة مع الطريقة الثانية، ومعامل التغاير ت هاتان الطريقتان مع الطرق المعتمدة في اربع قد قورنقط من الطريقة الثانية البديلة. ل% ف0.0لها اكبر بمقدار
(.ACI 318M-11 ،ACI 318M-99 ،BS ،NZمدونات هي )، أعطت الطريقتان المقترحتان Mtest/Mrبالاعتماد على نسبة مقاومة الانثناء العملية/مقاومة الانثناء المحسوبة
% للطريقة البديلة. بينما كانت قيم معامل التغاير 10.1 % للطريقة المبسطة،10.4قيما لمعامل التغاير قليلة نسبيا5 % للمدونات الاربع السابقة على التوالي.10.2% و %10.1، %10.4، 12.0
131( وهي ان جميع النماذج )=0.30هنالك فائدة أساسية عند تطبيق طريقة التصميم البسيطة )باستعمال قيمة ثابتة نموذجا ( أعطت تقديرا أمينا.
NUCEJ Vol.18 No.1. 2015 Khalel, et al., pp.1-15
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Details of 181 beam tests [6-28]
Beam No. Reference No. (MPa) b/d c/d
1 6 21.1 0.008804 1.000000 0.170137
2 6 21.1 0.017608 1.000000 0.340274
3 6 32.8 0.008804 1.000000 0.129649
4 6 32.8 0.012276 1.000000 0.163093
5 6 40.5 0.008804 1.000000 0.119597
6 6 40.5 0.012276 1.000000 0.150448
7 7 21.7 0.035031 0.585586 0.468651
8 7 30.8 0.008294 0.576923 0.089643
9 7 38.4 0.018963 0.565217 0.186527
10 7 25.0 0.017912 0.572072 0.217703
11 7 15.2 0.022691 0.822581 0.433382
12 7 36.2 0.018912 0.556034 0.459984
13 7 25.9 0.008273 0.542017 0.098417
14 7 36.2 0.008403 0.533613 0.084463
15 7 10.3 0.017712 0.545852 0.504922
16 7 21.8 0.033821 0.547009 0.455539
17 7 8.6 0.013236 0.547414 0.446816
18 7 24.9 0.034277 0.560870 0.422427
19 7 32.3 0.034210 0.560870 0.361255
20 7 21.4 0.019245 0.560870 0.264061
21 8 12.3 0.011516 0.809524 0.261420
22 8 10.8 0.015497 0.822581 0.435626
23 8 13.3 0.022876 0.772727 0.583175
24 8 13.6 0.030361 0.822581 0.684397
25 8 20.5 0.010907 0.776119 0.148561
26 8 19.0 0.015012 0.796875 0.239881
27 8 19.1 0.023031 0.809524 0.394151
28 8 19.0 0.023719 0.822581 0.376242
29 8 19.1 0.030794 0.793651 0.552000
30 8 19.0 0.030003 0.868852 0.491668
31 8 19.1 0.039392 0.838710 0.642147
32 8 31.0 0.011161 0.784615 0.122082
33 8 31.0 0.014781 0.784615 0.166538
34 8 27.2 0.022978 0.796875 0.304129
35 8 31.0 0.023263 0.838710 0.241056
36 8 27.2 0.030190 0.809524 0.403579
37 8 31.0 0.031501 0.836066 0.354918
38 8 35.7 0.011701 0.822581 0.120068
39 8 35.7 0.015813 0.822581 0.167134
40 8 35.7 0.023087 0.822581 0.267171
41 8 35.3 0.024510 0.850000 0.239350
42 8 34.2 0.029105 0.796875 0.346348
43 8 35.7 0.031501 0.836066 0.313511
44 8 30.7 0.037587 0.787879 0.444205
45 8 35.7 0.045906 0.838710 0.531245
46 8 12.3 0.005189 0.825397 0.187922
47 8 14.3 0.009615 0.812500 0.441813
48 8 19.0 0.005376 0.822581 0.135134
NUCEJ Vol.18 No.1. 2015 Khalel, et al., pp.1-15
11
Beam No. Reference No. (MPa) b/d c/d
49 8 22.1 0.009365 0.761194 0.270858
50 8 31.0 0.005376 0.822581 0.097494
51 8 28.6 0.009050 0.764706 0.219322
52 8 28.6 0.015152 0.772727 0.365248
53 8 30.7 0.009227 0.750000 0.226100
54 8 30.7 0.014277 0.787879 0.358107
55 8 32.9 0.023226 0.806452 0.415793
56 9 37.2 0.012035 0.720379 0.135262
57 9 36.2 0.018739 0.720379 0.239391
58 9 36.5 0.036169 0.720379 0.424616
59 9 15.4 0.021801 0.671053 0.583510
60 10 27.4 0.019100 0.541333 0.255942
61 10 28.2 0.019894 0.535809 0.214995
62 10 26.9 0.012797 0.544236 0.200585
63 10 32.6 0.012603 0.546917 0.158307
64 10 31.3 0.007906 0.548387 0.107097
65 10 29.8 0.007903 0.545455 0.125022
66 10 23.0 0.004404 0.545455 0.066124
67 10 25.0 0.004506 0.549865 0.077963
68 10 23.3 0.017810 1.201183 0.299721
69 10 28.7 0.017697 1.209581 0.282607
70 10 20.7 0.011106 0.749077 0.194404
71 10 27.6 0.010899 0.743590 0.168330
72 10 17.8 0.006204 0.424686 0.136326
73 10 23.3 0.006303 0.428571 0.112178
74 11 31.0 0.021474 0.769231 0.509683
75 11 29.5 0.021308 0.759494 0.512665
76 11 38.0 0.021807 0.779221 0.476576
77 12 27.2 0.011967 0.687783 0.236534
78 13 35.9 0.006528 0.584615 0.104026
79 13 37.3 0.014446 0.646809 0.226847
80 14 35.9 0.017407 0.740741 0.357108
81 14 39.4 0.017296 0.740741 0.343182
82 14 37.5 0.017500 0.740741 0.353114
83 14 39.3 0.017407 0.740741 0.345667
84 14 39.0 0.017500 0.740741 0.348370
85 14 39.9 0.017407 0.740741 0.344028
86 14 39.4 0.017407 0.740741 0.345387
87 15 27.6 0.014311 0.500000 0.259228
88 15 27.5 0.014311 0.500000 0.259763
89 15 27.4 0.014311 0.500000 0.260302
90 15 27.2 0.014311 0.500000 0.261396
91 15 26.7 0.014311 0.500000 0.264220
92 15 26.4 0.014311 0.500000 0.265977
93 15 25.7 0.014311 0.500000 0.270271
94 15 21.0 0.014311 0.500000 0.312613
95 15 27.1 0.007111 0.500000 0.130162
96 15 26.9 0.007111 0.500000 0.130720
97 15 26.4 0.007111 0.500000 0.132162
98 15 26.4 0.007111 0.500000 0.132162
NUCEJ Vol.18 No.1. 2015 Khalel, et al., pp.1-15
11
Beam No. Reference No. MPa) b/d c/d
99 15 26.1 0.007111 0.500000 0.133060
100 15 25.9 0.007111 0.500000 0.133672
101 15 25.5 0.007111 0.500000 0.134932
102 15 25.1 0.007111 0.500000 0.136239
103 15 24.7 0.007111 0.500000 0.137597
104 15 24.4 0.007111 0.500000 0.138649
105 15 21.0 0.007111 0.500000 0.155336
106 15 27.3 0.005711 0.500000 0.104096
107 15 27.2 0.005711 0.500000 0.104315
108 15 27.0 0.005711 0.500000 0.104759
109 15 27.0 0.005711 0.500000 0.104759
110 15 26.8 0.005711 0.500000 0.105212
111 15 26.6 0.005711 0.500000 0.105673
112 15 26.4 0.005711 0.500000 0.106143
113 15 26.2 0.005711 0.500000 0.106621
114 15 26.1 0.005711 0.500000 0.106864
115 15 25.8 0.005711 0.500000 0.107605
116 15 25.5 0.005711 0.500000 0.108367
117 15 25.2 0.005711 0.500000 0.109151
118 16 21.0 0.014505 1.342105 0.316853
119 16 41.0 0.014505 1.342105 0.220896
120 17 42.0 0.016318 1.490196 0.246903
121 17 42.0 0.016318 1.490196 0.246903
122 17 42.0 0.016318 1.490196 0.246903
123 17 42.0 0.016318 1.490196 0.246903
124 17 21.0 0.016318 1.490196 0.356460
125 17 21.0 0.016318 1.490196 0.356460
126 17 21.0 0.016318 1.490196 0.356460
127 18 32.7 0.013076 1.200787 0.216936
128 18 32.7 0.013076 1.200787 0.216936
129 19 45.8 0.009181 1.315789 0.171337
130 19 36.3 0.013333 1.327434 0.301466
131 19 46.5 0.010572 1.515152 0.196843
132 19 46.2 0.015374 1.530612 0.325087
133 20 21.0 0.030000 0.716418 0.609216
134 20 110.0 0.030000 0.716418 0.220606
135 21 30.0 0.018985 0.597015 0.396915
136 21 30.0 0.005866 0.597015 0.122632
137 22 85.0 0.068447 0.851064 0.591992
138 22 85.0 0.068447 0.851064 0.591992
139 22 85.0 0.068447 0.851064 0.591992
140 22 85.0 0.038511 0.851064 0.402726
141 23 40.0 0.019286 0.666667 0.310832
142 23 60.0 0.019286 0.666667 0.248571
143 23 75.0 0.025850 0.666667 0.275737
144 23 40.0 0.002093 0.833333 0.030636
145 23 40.0 0.003533 0.833333 0.054067
146 23 40.0 0.019286 0.666667 0.310832
147 23 60.0 0.019286 0.666667 0.248571
148 23 75.0 0.003533 0.833333 0.034590
149 23 75.0 0.003533 0.833333 0.034590
150 23 75.0 0.025850 0.666667 0.275737
151 23 40.0 0.013673 0.666667 0.211766
152 23 40.0 0.019286 0.666667 0.310832
153 23 40.0 0.025850 0.666667 0.431002
154 23 40.0 0.041538 0.717949 0.604404
NUCEJ Vol.18 No.1. 2015 Khalel, et al., pp.1-15
12
Beam No. Reference No. (MPa) b/d c/d
155 23 60.0 0.019286 0.666667 0.248571
156 23 71.0 0.019286 0.666667 0.210060
157 23 71.0 0.025850 0.666667 0.291271
158 23 71.0 0.041538 0.717949 0.452438
159 23 75.0 0.055714 0.717949 0.578154
160 24 31.0 0.005183 0.917431 0.088455
161 25 41.2 0.019286 0.666667 0.308247
162 25 41.0 0.019286 0.666667 0.308665
163 25 41.2 0.025850 0.666667 0.427415
164 25 41.2 0.025850 0.666667 0.427415
165 25 41.0 0.041538 0.717949 0.603166
166 25 41.2 0.041538 0.717949 0.602934
167 25 71.2 0.025850 0.666667 0.290518
168 25 71.2 0.025850 0.666667 0.290518
169 25 71.2 0.041538 0.717949 0.451271
170 25 71.2 0.041538 0.717949 0.451271
171 26 35.0 0.008619 0.746269 0.140784
172 27 32.3 0.003379 0.833333 0.052429
173 27 32.3 0.004180 0.833333 0.062296
174 27 32.3 0.003971 0.833333 0.052978
175 28 36.75 0.002778 0.555556 0.051279
176 28 36.75 0.002778 0.555556 0.051279
177 28 36.75 0.002778 0.555556 0.051279
178 28 36.75 0.005611 0.555556 0.103584
179 28 36.75 0.005611 0.555556 0.103584
180 28 36.75 0.005611 0.555556 0.103584
181 28 36.75 0.005611 0.555556 0.103584
Ratio of ( ) for 181 beam tests6-28
Beam
No. Reference No. ACI-11
1 ACI-99
3 BS-97
4 NZ-95
5
Simple
Method
Alternative
Method
1 6 1.01042 1.01042 1.00051 1.01042 1.06986 0.99997
2 6 1.20214 1.09729 1.14996 1.09729 1.16183 1.07240
3 6 1.10917 1.10917 1.07981 1.10917 1.17442 1.10738
4 6 1.13290 1.13290 1.11145 1.13290 1.19954 1.13056
5 6 1.08226 1.08226 1.04784 1.08226 1.14592 1.08436
6 6 1.06665 1.06665 1.03899 1.06665 1.12940 1.06930
7 7 1.59196 1.16812 1.33584 1.16812 1.23683 1.23879
8 7 1.19078 1.19078 1.14979 1.19078 1.26082 1.18862
9 7 1.14149 1.14149 1.12165 1.14149 1.20863 1.14343
10 7 1.21106 1.21106 1.21290 1.21106 1.28230 1.19848
11 7 1.44681 1.11748 1.22130 1.11748 1.18322 1.15240
12 7 1.37937 1.13512 1.29478 1.13585 1.20949 1.23718
13 7 1.40044 1.40044 1.35735 1.40044 1.48282 1.39479
14 7 1.29869 1.29869 1.25055 1.29869 1.37509 1.29877
15 7 1.62112 1.17081 1.39669 1.17081 1.23968 1.27292
16 7 1.55918 1.16536 1.31284 1.16536 1.23391 1.22365
17 7 1.77046 1.34015 1.49114 1.34015 1.41898 1.39750
18 7 1.37965 1.11671 1.20083 1.11671 1.18240 1.14858
19 7 1.18245 1.13086 1.17082 1.13086 1.19739 1.12425
20 7 1.14142 1.14142 1.16401 1.14142 1.20857 1.12216
21 8 1.29585 1.29585 1.32032 1.29585 1.37208 1.27423
22 8 1.66930 1.28492 1.41557 1.28492 1.36050 1.32759
23 8 1.52295 1.09991 1.37184 1.09991 1.16461 1.19810
NUCEJ Vol.18 No.1. 2015 Khalel, et al., pp.1-15
13
Beam
No. Reference No. ACI-11
1 ACI-99
3 BS-97
4 NZ-95
5
Simple
Method
Alternative
Method
24 8 1.98160 1.43115 1.80621 1.43115 1.51534 1.48753
25 8 1.09045 1.09045 1.07307 1.09045 1.15459 1.08071
26 8 1.12617 1.12617 1.13931 1.12617 1.19241 1.10913
27 8 1.47650 1.21582 1.30188 1.21582 1.28734 1.20898
28 8 1.48466 1.26242 1.34185 1.26242 1.33668 1.23191
29 8 1.69211 1.22208 1.51953 1.22208 1.29397 1.33931
30 8 1.64027 1.18464 1.40135 1.18464 1.25433 1.27683
31 8 1.87108 1.35134 1.69953 1.35134 1.43083 1.43937
32 8 1.19635 1.19635 1.16364 1.19635 1.26672 1.19347
33 8 1.02060 1.02060 1.00327 1.02060 1.08064 1.01719
34 8 1.11535 1.11535 1.14394 1.11535 1.18096 1.10205
35 8 1.08823 1.08823 1.09039 1.08823 1.15224 1.08276
36 8 1.37199 1.17445 1.24653 1.17445 1.24353 1.19237
37 8 1.23971 1.19247 1.23507 1.19247 1.26262 1.18313
38 8 1.28594 1.28594 1.24764 1.28594 1.36159 1.28577
39 8 1.12694 1.12694 1.10462 1.12694 1.19323 1.12673
40 8 1.19162 1.19162 1.19601 1.19162 1.26171 1.19123
41 8 1.14684 1.14684 1.14363 1.14684 1.21431 1.14608
42 8 1.14811 1.14467 1.17601 1.14467 1.21200 1.14163
43 8 1.16453 1.16453 1.18290 1.16453 1.23303 1.16407
44 8 1.43775 1.17809 1.25825 1.17809 1.24739 1.25189
45 8 1.51203 1.12775 1.24179 1.12775 1.19408 1.29637
46 8 1.23269 1.23269 1.22704 1.23269 1.30520 1.21848
47 8 1.55543 1.23653 1.47086 1.23653 1.30927 1.22891
48 8 1.13302 1.13302 1.11078 1.13302 1.19967 1.12388
49 8 1.15562 1.15562 1.18104 1.15562 1.22360 1.13571
50 8 1.17644 1.17644 1.13787 1.17644 1.24564 1.17421
51 8 1.11096 1.11096 1.10927 1.11096 1.17631 1.10337
52 8 1.30939 1.21765 1.27536 1.21765 1.28927 1.20270
53 8 1.13508 1.13508 1.13314 1.13508 1.20185 1.12944
54 8 1.27339 1.21491 1.26027 1.21491 1.28638 1.20468
55 8 1.44743 1.25594 1.32138 1.25594 1.32982 1.30563
56 9 1.23614 1.23614 1.20235 1.23614 1.30886 1.23690
57 9 1.22103 1.22103 1.21656 1.22103 1.29285 1.22126
58 9 1.33278 1.16199 1.21630 1.16199 1.23034 1.22677
59 9 1.70973 1.23481 1.52568 1.23481 1.30744 1.30514
60 10 1.02327 1.02327 1.03378 1.02327 1.08346 1.01351
61 10 1.12869 1.12869 1.12606 1.12869 1.19508 1.12071
62 10 0.95376 0.95376 0.94886 0.95376 1.00986 0.94639
63 10 1.06845 1.06845 1.04717 1.06845 1.13130 1.06618
64 10 1.10723 1.10723 1.07313 1.10723 1.17236 1.10505
65 10 1.11082 1.11082 1.08180 1.11082 1.17616 1.10739
66 10 1.21717 1.21717 1.17138 1.21717 1.28877 1.21289
67 10 1.10320 1.10320 1.06422 1.10320 1.16810 1.09937
68 10 0.98470 0.98470 1.01456 0.98470 1.04262 0.96745
69 10 1.17481 1.17481 1.19438 1.17481 1.24392 1.16429
70 10 1.14413 1.14413 1.14111 1.14413 1.21143 1.13044
71 10 1.14367 1.14367 1.12734 1.14367 1.21095 1.13698
72 10 1.17919 1.17919 1.15643 1.17919 1.24856 1.16959
73 10 1.10373 1.10373 1.07467 1.10373 1.16866 1.09717
74 11 1.38518 1.07903 1.28023 1.07956 1.15206 1.15668
75 11 1.38519 1.07473 1.28311 1.07473 1.14495 1.14195
76 11 1.30489 1.05898 1.18966 1.05898 1.12127 1.17468
77 12 1.02501 1.02501 1.02990 1.02501 1.08531 1.01584
NUCEJ Vol.18 No.1. 2015 Khalel, et al., pp.1-15
14
Beam
No. Reference No. ACI-11
1 ACI-99
3 BS-97
4 NZ-95
5
Simple
Method
Alternative
Method
78 13 1.24753 1.24753 1.20622 1.24753 1.32092 1.24748
79 13 1.22189 1.22189 1.21258 1.22189 1.29377 1.22330
80 14 1.15831 1.14007 1.17155 1.14007 1.20714 1.13990
81 14 1.12825 1.12825 1.14873 1.12825 1.19462 1.13357
82 14 1.12645 1.12645 1.15326 1.12645 1.19271 1.12892
83 14 1.04158 1.04158 1.06131 1.04158 1.10285 1.04640
84 14 1.03826 1.03826 1.05914 1.03826 1.09933 1.04267
85 14 1.03906 1.03906 1.05735 1.03906 1.10019 1.04467
86 14 1.04116 1.04116 1.06064 1.04116 1.10240 1.04611
87 15 1.27416 1.27416 1.28819 1.27416 1.34911 1.26213
88 15 1.29726 1.29726 1.31191 1.29726 1.37357 1.28483
89 15 1.30813 1.30813 1.32328 1.30813 1.38508 1.29541
90 15 1.32005 1.32005 1.33610 1.32005 1.39770 1.30683
91 15 1.33856 1.33856 1.35683 1.33856 1.41729 1.32415
92 15 1.34676 1.34676 1.36640 1.34676 1.42599 1.33165
93 15 1.36426 1.36426 1.38725 1.36426 1.44451 1.34742
94 15 1.35136 1.31164 1.36052 1.31164 1.38879 1.28473
95 15 1.51805 1.51805 1.48222 1.51805 1.60734 1.51087
96 15 1.52390 1.52390 1.48830 1.52390 1.61354 1.51648
97 15 1.53157 1.53157 1.49673 1.53157 1.62166 1.52359
98 15 1.54025 1.54025 1.50521 1.54025 1.63085 1.53222
99 15 1.54987 1.54987 1.51520 1.54987 1.64104 1.54146
100 15 1.55342 1.55342 1.51907 1.55342 1.64480 1.54476
101 15 1.56057 1.56057 1.52689 1.56057 1.65237 1.55140
102 15 1.56809 1.56809 1.53511 1.56809 1.66033 1.55840
103 15 1.57569 1.57569 1.54344 1.57569 1.66837 1.56545
104 15 1.57986 1.57986 1.54822 1.57986 1.67279 1.56921
105 15 1.56856 1.56856 1.54655 1.56856 1.66083 1.55386
106 15 1.57057 1.57057 1.52353 1.57057 1.66296 1.56485
107 15 1.58079 1.58079 1.53359 1.58079 1.67378 1.57495
108 15 1.58839 1.58839 1.54125 1.58839 1.68183 1.58235
109 15 1.58839 1.58839 1.54125 1.58839 1.68183 1.58235
110 15 1.59601 1.59601 1.54894 1.59601 1.68990 1.58977
111 15 1.60365 1.60365 1.55665 1.60365 1.69798 1.59720
112 15 1.61130 1.61130 1.56439 1.61130 1.70608 1.60464
113 15 1.61824 1.61824 1.57144 1.61824 1.71343 1.61136
114 15 1.61527 1.61527 1.56872 1.61527 1.71029 1.60832
115 15 1.61964 1.61964 1.57345 1.61964 1.71492 1.61239
116 15 1.62773 1.62773 1.58180 1.62773 1.72347 1.62015
117 15 1.63253 1.63253 1.58698 1.63253 1.72856 1.62463
118 16 1.18729 1.14109 1.18545 1.14109 1.20821 1.11730
119 16 1.22192 1.22192 1.20718 1.22192 1.29379 1.22699
120 17 1.42438 1.42438 1.41377 1.42438 1.50817 1.43233
121 17 1.16018 1.16018 1.15154 1.16018 1.22843 1.16666
122 17 1.38992 1.38992 1.37956 1.38992 1.47168 1.39768
123 17 1.37844 1.37844 1.36816 1.37844 1.45952 1.38613
124 17 1.67043 1.47517 1.55568 1.47517 1.56194 1.43979
125 17 1.48160 1.30841 1.37982 1.30841 1.38537 1.27703
126 17 1.48160 1.30841 1.37982 1.30841 1.38537 1.27703
127 18 1.57625 1.57625 1.56692 1.57625 1.66897 1.57167
128 18 1.57740 1.57740 1.56807 1.57740 1.67019 1.57282
129 19 1.48042 1.48042 1.44357 1.48042 1.56750 1.48915
130 19 1.25085 1.25085 1.26555 1.25085 1.32443 1.25131
131 19 1.73753 1.73753 1.70147 1.73753 1.83974 1.75019
NUCEJ Vol.18 No.1. 2015 Khalel, et al., pp.1-15
15
Beam
No. Reference No. ACI-11
1 ACI-99
3 BS-97
4 NZ-95
5
Simple
Method
Alternative
Method
132 19 1.26126 1.26126 1.26693 1.26126 1.33545 1.27689
133 20 2.32595 1.67985 2.07649 1.67985 1.77867 1.74872
134 20 1.20105 1.20105 1.17825 1.21379 1.27170 1.21379
135 21 1.19434 1.05444 1.11837 1.05444 1.11647 1.06923
136 21 0.97121 0.97121 0.94521 0.97121 1.02834 0.96837
137 22 1.35227 0.98183 1.02781 1.08341 1.03958 1.20061
138 22 1.33181 0.96697 1.01226 1.06701 1.02385 1.18244
139 22 1.36457 0.99076 1.03716 1.09326 1.04904 1.21153
140 22 1.11514 1.04460 1.06064 1.06646 1.10604 1.10012
141 23 1.26613 1.26613 1.27781 1.26613 1.34060 1.27235
142 23 1.29167 1.29167 1.27339 1.30729 1.36765 1.30729
143 23 1.24684 1.24684 1.23523 1.26375 1.32018 1.26375
144 23 1.79318 1.79318 1.70796 1.79318 1.89866 1.79395
145 23 1.51058 1.51058 1.44488 1.51058 1.59944 1.51174
146 23 1.25340 1.25340 1.26497 1.25340 1.32713 1.25957
147 23 1.30756 1.30756 1.28906 1.32338 1.38448 1.32338
148 23 1.60380 1.60380 1.52709 1.60628 1.69814 1.60628
149 23 2.31515 2.31515 2.20442 2.31873 2.45133 2.31873
150 23 1.20870 1.20870 1.19745 1.22510 1.27980 1.22510
151 23 1.12171 1.12171 1.10692 1.12171 1.18770 1.12531
152 23 1.20862 1.20862 1.21978 1.20862 1.27971 1.21456
153 23 1.35960 1.19614 1.24511 1.19614 1.26651 1.27863
154 23a 1.87927 1.34478 1.56786 1.35221 1.43709 1.61613
155 23 1.32720 1.32720 1.30842 1.34325 1.40527 1.34325
156 23 1.30626 1.30626 1.27912 1.31940 1.38310 1.31940
157 23 1.20017 1.20017 1.19239 1.21748 1.27077 1.21748
158 23 1.35516 1.16856 1.19900 1.19665 1.23730 1.29517
159 23 1.54124 1.13616 1.18417 1.25256 1.20299 1.37691
160 24 1.47305 1.47305 1.42187 1.47305 1.55970 1.47053
161 25 1.25973 1.25973 1.26864 1.25973 1.33383 1.26762
162 25 1.20425 1.20425 1.21319 1.20425 1.27508 1.21152
163 25 1.29995 1.15837 1.20167 1.15837 1.22651 1.23640
164 25 1.32535 1.18100 1.22514 1.18100 1.25047 1.26055
165 25 1.85080 1.32423 1.53479 1.33240 1.41532 1.60191
166 25 1.54960 1.10869 1.28346 1.11568 1.18499 1.34294
167 25 1.21035 1.21035 1.20234 1.22776 1.28155 1.22776
168 25 1.19957 1.19957 1.19163 1.21683 1.27013 1.21683
169 25 1.35100 1.16708 1.19718 1.19505 1.23573 1.29211
170 25 1.43020 1.23550 1.26736 1.26511 1.30818 1.36786
171 26 1.68241 1.68241 1.64014 1.68241 1.78137 1.68150
172 27 1.06000 1.06000 1.01494 1.06000 1.12236 1.05923
173 27 1.30851 1.30851 1.25548 1.30851 1.38548 1.30737
174 27 1.32941 1.32941 1.27304 1.32941 1.40762 1.32843
175 28 0.97757 0.97757 0.93508 0.97757 1.03508 0.97771
176 28 1.14180 1.14180 1.09217 1.14180 1.20897 1.14196
177 28 1.36860 1.36860 1.30911 1.36860 1.44911 1.36879
178 28 1.30694 1.30694 1.26315 1.30694 1.38382 1.30731
179 28 1.38615 1.38615 1.33971 1.38615 1.46768 1.38655
180 28 1.38615 1.38615 1.33971 1.38615 1.46768 1.38655
181 28 1.38615 1.38615 1.33971 1.38615 1.46768 1.38655